Euler's number

This article covers the base of the natural logarithm. For other numbers named after Euler see Euler's numbers (disambiguation) .

The Euler's number , with the symbol referred to, is a constant that throughout the analysis and all related areas of mathematics , especially in the differential and integral calculus , plays a central role. Its numerical value is ${\displaystyle e}$

${\displaystyle e=2{,}71828\,18284\,59045\,23536\,02874\,71352\,66249\,77572\,47093\,69995\,\dots }$

${\displaystyle e}$is a transcendent and therefore also an irrational real number . It is the basis of the natural logarithm and the (natural) exponential function . In applied mathematics , the exponential function plays an important role in describing processes such as radioactive decay and natural growth . There are numerous equivalent definitions of , the most popular of which is: ${\displaystyle e}$ ${\displaystyle e}$

${\displaystyle e=1+{\frac {1}{1}}+{\frac {1}{1\cdot 2}}+{\frac {1}{1\cdot 2\cdot 3}}+{\frac {1}{1\cdot 2\cdot 3\cdot 4}}+\dotsb =\sum _{k=0}^{\infty }{\frac {1}{k!}}}$

The number was named after the Swiss mathematician Leonhard Euler , who described numerous properties of . It is sometimes referred to as Napier's constant after the Scottish mathematician John Napier . It is one of the most important constants in mathematics. ${\displaystyle e}$

definition

The number was defined by Leonhard Euler by the following series : ${\displaystyle e}$

${\displaystyle {\begin{aligned}e&=1+{\frac {1}{1}}+{\frac {1}{1\cdot 2}}+{\frac {1}{1\cdot 2\cdot 3}}+{\frac {1}{1\cdot 2\cdot 3\cdot 4}}+\dotsb \\&={\frac {1}{0!}}+{\frac {1}{1!}}+{\frac {1}{2!}}+{\frac {1}{3!}}+{\frac {1}{4!}}+\dotsb \\&=\sum _{k=0}^{\infty }{\frac {1}{k!}}\\\end{aligned}}}$

For is the factorial of , i.e. in the case of the product of the natural numbers from to , while is defined. ${\displaystyle k\in \mathbb {N} _{0}}$ ${\displaystyle k!}$ ${\displaystyle k}$ ${\displaystyle k>0}$ ${\displaystyle k!=1\cdot 2\cdot \ldots \cdot k}$ ${\displaystyle 1}$ ${\displaystyle k}$ ${\displaystyle 0!=1}$

As Euler already proved, Euler's number is also obtained as a functional limit value : ${\displaystyle e}$

${\displaystyle e=\lim _{t\to \infty \atop t\in \mathbb {R} }\left(1+{\frac {1}{t}}\right)^{t}}$,

which means in particular that he is also known as limit the result with results: ${\displaystyle (a_{n})_{n\in \mathbb {N} }}$ ${\displaystyle a_{n}:=\left(1+{\frac {1}{n}}\right)^{n}}$

${\displaystyle e=\lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n}}$.

This is based on the fact that

${\displaystyle e=\exp(1)=e^{1}}$

applies, i.e. the function value of the exponential function (or " function") is at that point . The above series representation of results in this context by evaluating the Taylor series of the exponential function around the development point at the point . ${\displaystyle e}$ ${\displaystyle e}$ ${\displaystyle 1}$ ${\displaystyle e}$ ${\displaystyle 0}$ ${\displaystyle 1}$

An alternative approach to the definition of the Euler number is the one via of intervals , such as in the manner as described in Theory and Application of infinite series of Konrad Knopp is illustrated. After that applies to everyone : ${\displaystyle n\in \mathbb {N} }$

${\displaystyle \left(1+{\frac {1}{n}}\right)^{n}<e<\left(1+{\frac {1}{n}}\right)^{n+1}}$.

Origin of the symbol e

The earliest document showing the use of the letter for this number by Leonhard Euler is a letter from Euler to Christian Goldbach dated November 25, 1731. The closest reliable source for the use of this letter is Euler's work Mechanica sive motus scientia analytice exposita, II from 1736. ${\displaystyle e}$

In the Introductio in Analysin Infinitorum , published in 1748 , Euler takes up this name again.

There is no evidence that this choice of letter was based on his name. It is also unclear whether he did this on the basis of the exponential function or for practical considerations of delimitation from the frequently used letters a, b, c or d . Although other names were also in use, such as c in d'Alembert's Histoire de l'Académie, has become established. ${\displaystyle e}$ ${\displaystyle e}$

In accordance with DIN 1338 and ISO 80000-2, italics are not used in the formula set in order to distinguish the number from a variable. However, italic notation is also common. ${\displaystyle e}$

properties

Euler's number is a transcendent ( proof according to Charles Hermite , 1873) and thus an irrational number (proof with continued fractions for and thus by Euler as early as 1737, proof ). It can therefore (like the circle number after Ferdinand von Lindemann 1882) neither be represented as a fraction of two natural numbers nor as a solution of an algebraic equation and consequently has an infinite non-periodic decimal fraction expansion . The measure of irrationality of is 2 and therefore as small as possible for an irrational number, in particular it is not liouville . It is not known if is normal on any basis . ${\displaystyle e}$ ${\displaystyle e^{2}}$ ${\displaystyle e}$ ${\displaystyle \pi }$ ${\displaystyle e}$ ${\displaystyle e}$ ${\displaystyle e}$

In Euler's identity

${\displaystyle e^{\mathrm {i} \cdot \pi }=-1}$

fundamental mathematical constants are put in context: the integer 1, Euler's number , the imaginary unit of complex numbers and the circle number . ${\displaystyle e}$ ${\displaystyle \mathrm {i} }$ ${\displaystyle \pi }$

Euler's number also appears in the asymptotic estimate of the faculty (see Stirling's formula ):

${\displaystyle {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}\leq n!\leq {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}\cdot e^{\frac {1}{12n}}}$

With the Cauchy product formula of the absolutely convergent series, the following applies :

${\displaystyle \sum _{k=0}^{\infty }{\frac {1}{k!}}\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{k!}}}$ with the product line . ${\displaystyle \sum _{k=0}^{\infty }{0^{k}}=e{\frac {1}{e}}=1}$

Geometric interpretation

The integral calculus provides a geometric interpretation of Euler's number . According to this, the uniquely determined number is for which the content of the area below the function graph of the real reciprocal function in the interval is exactly the same : ${\displaystyle e}$ ${\displaystyle b>1}$ ${\displaystyle y={\tfrac {1}{x}}}$ ${\displaystyle [1,b]}$ ${\displaystyle 1}$

${\displaystyle \int _{1}^{e}{\frac {1}{x}}\,\mathrm {d} x=1}$

Further representations for Euler's number

Euler's number can also be passed through

${\displaystyle e=\lim _{n\to \infty }{\frac {n}{\sqrt[{n}]{n!}}}}$

or by the limit of the quotient of the faculty and the sub-faculty :

${\displaystyle e=\lim _{n\to \infty }{\frac {n!}{!n}}.}$

A connection to the distribution of prime numbers is made via the formulas

${\displaystyle e=\lim _{n\to \infty }({\sqrt[{n}]{n}})^{\pi (n)}}$

${\displaystyle e=\lim _{n\to \infty }{\sqrt[{n}]{n\#}}}$

clearly, where the prime number function and the symbol mean the primorial of the number . ${\displaystyle \pi (n)}$ ${\displaystyle n\#}$ ${\displaystyle n}$

The Catalan representation is also more exotic than of practical importance

${\displaystyle e={\sqrt[{1}]{\frac {2}{1}}}\cdot {\sqrt[{2}]{\frac {4}{3}}}\cdot {\sqrt[{4}]{\frac {6\cdot 8}{5\cdot 7}}}\cdot {\sqrt[{8}]{\frac {10\cdot 12\cdot 14\cdot 16}{9\cdot 11\cdot 13\cdot 15}}}\cdots }$

Continued fractions

In connection with the number , at least since the appearance of Leonhard Euler's Introductio in Analysin Infinitorum in 1748, there has been a large number of continued fraction developments for and from derivable quantities. ${\displaystyle e}$ ${\displaystyle e}$ ${\displaystyle e}$

So Euler found the following classic identity for : ${\displaystyle e}$

${\displaystyle (1){\begin{aligned}e&=[2;1,2,1,1,4,1,1,6,1,1,8,1,1,10,1,\dotsc ]\\&=2+{\cfrac {1}{1+{\cfrac {1}{2+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{4+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{6+\dotsb }}}}}}}}}}}}}}}}\end{aligned}}}$(Follow A003417 in OEIS )

Identity (1) evidently shows a regular pattern that continues indefinitely. It represents a regular continued fraction which Euler derived from the following:

${\displaystyle (2){\begin{aligned}{\frac {e+1}{e-1}}&=[2;6,10,14,\dotsc ]\\&={2+{\cfrac {1}{6+{\cfrac {1}{10+{\cfrac {1}{14+{\cfrac {1}{\;\,\ddots }}}}}}}}}\\&\approx 2{,}1639534137386\end{aligned}}}$(Follow A016825 in OEIS )

The latter continued fraction is in turn a special case of the following with : ${\displaystyle k=2}$

${\displaystyle (3){\begin{aligned}{\coth {\frac {1}{k}}}&={\frac {e^{\frac {2}{k}}+1}{e^{\frac {2}{k}}-1}}\\&=[k;3k,5k,7k,\dots ]\\&={k+{\cfrac {1}{3k+{\cfrac {1}{5k+{\cfrac {1}{7k+{\cfrac {1}{\;\,\ddots }}}}}}}}}\\\end{aligned}}}$     ${\displaystyle (k=1,2,3,\dots )}$

Another classic continued fraction expansion, which however is not regular , also comes from Euler:

${\displaystyle (4){\frac {1}{e-1}}={0+{\cfrac {1}{1+{\cfrac {2}{2+{\cfrac {3}{3+{\cfrac {4}{\;\,\ddots }}}}}}}}}\approx 0{,}58197670686932}$

Another continued fraction expansion of Euler's number goes back to Euler and Ernesto Cesàro , which has a different pattern than in (1):

${\displaystyle (5){\begin{aligned}e&=2+{\cfrac {1}{1+{\cfrac {1}{2+{\cfrac {2}{3+{\cfrac {3}{4+{\cfrac {4}{5+{\cfrac {5}{6+{\cfrac {6}{7+{\cfrac {7}{8+\dotsb }}}}}}}}}}}}}}}}\end{aligned}}}$

In connection with Euler's number, there is also a large number of general chain break theoretical functional equations . Oskar Perron names the following generally applicable representation of the function as one of several : ${\displaystyle e}$

${\displaystyle (6){\begin{aligned}{e^{z}}&=1+{\cfrac {z}{1-{\cfrac {1z}{2+z-{\cfrac {2z}{3+z-{\cfrac {3z}{4+z-{\cfrac {4z}{5+z-{\cfrac {5z}{6+z-{\cfrac {6z}{7+z-{\cfrac {7z}{8+z-\dotsb }}}}}}}}}}}}}}}}\end{aligned}}}$     ${\displaystyle (z\in \mathbb {C} )}$

Another example of this is the development of the hyperbolic tangent from Johann Heinrich Lambert , which is included in Lambert's continued fractions :

${\displaystyle (7){\begin{aligned}{\tanh z}&={\frac {e^{z}-e^{-z}}{e^{z}+e^{-z}}}\\&={\frac {e^{2z}-1}{e^{2z}+1}}\\&=0+{\cfrac {z}{1+{\cfrac {z^{2}}{3+{\cfrac {z^{2}}{5+{\cfrac {z^{2}}{7+{\cfrac {z^{2}}{9+{\cfrac {z^{2}}{11+{\cfrac {z^{2}}{13+{\cfrac {z^{2}}{15+\dotsb }}}}}}}}}}}}}}}}\end{aligned}}}$     ${\displaystyle \left(z\in \mathbb {C} \setminus \left\{{\frac {\mathrm {i} \pi }{2}}+k\pi \colon k=0,1,2,3,\dotsc \right\}\right)}$

It was not until 2019 that a team led by Gal Raayoni at the Technion found another and previously unknown continued fraction expansion for Euler's number with the help of a computer program named after Srinivasa Ramanujan as the Ramanujan machine , ultimately based on a trial-and-error method . Compared to all previously known continued fraction expansions, which all rise from any integer number that is smaller than Euler's number, this is the first time that we are dealing with one from the integer 3 , an integer that is greater than Euler's number , descends. The mere finding of a (single) such descending continued fraction of an integer greater than Euler's number (3> e) suggests that there are infinitely many such descending continued fractions of integers n with n> e that also refer to the Lead Euler's number.

${\displaystyle (8){\begin{aligned}e&=3+{\cfrac {-1}{4+{\cfrac {-2}{5+{\cfrac {-3}{6+{\cfrac {-4}{7+{\cfrac {-5}{8+\dotsb }}}}}}}}}}\end{aligned}}}$

Illustrative interpretations of Euler's number

Compound interest calculation

The following example not only makes the calculation of Euler's number clearer, but it also describes the history of the discovery of Euler's number: Its first digits were found by Jakob I Bernoulli when he investigated the calculation of compound interest.

The limit value of the first formula can be interpreted as follows: someone deposits one euro in the bank on January 1st . The bank guarantees him a current interest rate at one rate per year. What is his balance on January 1st of the next year if he invests the interest on the same terms? ${\displaystyle z=100\,\%}$

After the compound interest is calculated from the starting capital for interest rates with interest rate capital ${\displaystyle K_{0}}$ ${\displaystyle n}$ ${\displaystyle z}$

${\displaystyle K_{n}=K_{0}(1+z)^{n}.}$

In this example, and if the interest is payable annually, or if the interest surcharge done times a year, ie in less than one year return . ${\displaystyle K_{0}=1}$ ${\displaystyle z=100\,\%=1}$ ${\displaystyle z=1/n}$ ${\displaystyle n}$

With an annual surcharge it would be

${\displaystyle K_{1}=1\cdot (1+1)^{1}=2{,}00.}$

In semi-annual award has , ${\displaystyle z={\frac {1}{2}}}$

${\displaystyle K_{2}=1\cdot \left(1+{\frac {1}{2}}\right)^{2}=2{,}25}$

so a little more. With daily interest you get ${\displaystyle \left(z={\frac {1}{365}}\right)}$

${\displaystyle K_{365}=1\cdot \left(1+{\frac {1}{365}}\right)^{365}=2{,}714567.}$

If the compounding occurs continuously at every instant, it becomes infinitely large, and you get the first formula for . ${\displaystyle n}$ ${\displaystyle e}$

probability calculation

${\displaystyle e}$is also often found in probability theory : For example, assume that a baker puts a raisin into the dough for each roll and kneads it well. According to this, statistically speaking, every -th bun does not contain a raisin. The probability that none of the raisins is in a firmly selected one for rolls results in the limit value for ( 37% rule ): ${\displaystyle e}$ ${\displaystyle p}$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle n\to \infty }$

${\displaystyle p=\lim _{n\to \infty }\left({\frac {n-1}{n}}\right)^{n}=\lim _{n\to \infty }\left(1-{\frac {1}{n}}\right)^{n}={\frac {1}{e}}.}$

Characterization of Euler's number according to Steiner

In the fortieth volume of Crelles Journal from 1850, the Swiss mathematician Jakob Steiner gives a characterization of Euler's number , according to which it can be understood as the solution to an extreme value problem . Steiner showed namely that the number can be characterized as that uniquely determined positive real number that the square root gives to itself the greatest root. Steiner writes literally: "If every number is radiated by itself, the number e grants the very greatest root." ${\displaystyle e}$ ${\displaystyle e}$ ${\displaystyle e}$

Steiner deals with the question of whether for the function

${\displaystyle f\colon (0,\infty )\to (0,\infty ),\;x\mapsto f(x)={\sqrt[{x}]{x}}=x^{\frac {1}{x}}}$

the global maximum exists and how to determine it. Its proposition is that it exists and that it is accepted in and only in . ${\displaystyle x_{\mathrm {max} }=e}$

In his book Triumph der Mathematik Heinrich Dörrie gives an elementary solution to this extreme value problem. His approach is based on the following true statement about the real exponential function :

${\displaystyle \forall y\in \mathbb {R} \setminus \{0\}\colon e^{y}>1+y}$

After the substitution follows for all real numbers ${\displaystyle y={\frac {x-e}{e}}}$ ${\displaystyle x\neq e}$

${\displaystyle e^{\frac {x-e}{e}}>1+{\frac {x-e}{e}},}$

by means of simple transformations further

${\displaystyle e^{\frac {x}{e}}>x}$

and finally for all positive ones by rooting ${\displaystyle x\neq e}$

${\displaystyle {\sqrt[{e}]{e}}>{\sqrt[{x}]{x}}.}$

Fracture approximations

There are various approximate representations using fractions for the number and the quantities derived from it . How Charles Hermite found the following fractional approximations: ${\displaystyle e}$

${\displaystyle e\approx {\frac {58291}{21444}}\approx 2{,}718289498}$

${\displaystyle e^{2}\approx {\frac {158452}{21444}}\approx 7{,}38910651}$

Here the first fraction differs by less than 0.0003 percent from. ${\displaystyle e}$

The optimal fractional approximation in the three-digit range, i.e. the optimal fractional approximation with , is ${\displaystyle e\approx {\frac {Z_{0}}{N_{0}}}}$ ${\displaystyle N_{0},Z_{0}<1000}$

${\displaystyle e\approx {\frac {878}{323}}\approx 2{,}718266254}$.

However, this approximation is not the best fractional approximation in terms of the requirement that the denominator should have a maximum of three digits. The best fractional approximation in this sense results as the 9th  approximate fraction of the continued fraction expansion of Euler's number:

${\displaystyle e\approx {\frac {1457}{536}}\approx 2{,}71828358\dots }$

From the approximate fractions of the associated continued fraction expansions (see above), fractional approximations of any precision for and from these derived quantities result. With these one finds the best fractional approximations of the Euler's number in any number ranges very efficiently . This gives the best fractional approximation in the five-digit number range ${\displaystyle e}$ ${\displaystyle e}$

${\displaystyle e\approx {\frac {49171}{18089}}\approx 2{,}718281828735\dots }$,

which shows that the fractional approximation found by Charles Hermite for Euler's number in the five-digit range was not yet optimal.

C. D. Olds has shown in the same way that by approximation

${\displaystyle {\frac {e-1}{2}}\approx {\frac {342762}{398959}}}$

a further improvement for Euler's number, namely

${\displaystyle e\approx {\frac {1084483}{398959}}\approx 2{,}7182818284585\dots }$,

can be achieved.

Overall, the sequence of the best approximate fractions of Euler's number, which result from its regular continued fraction representation, begins as follows:

${\displaystyle {\frac {p_{0}}{q_{0}}}=[2]={\frac {2}{1}}}$

${\displaystyle {\frac {p_{1}}{q_{1}}}=[2;1]={\frac {3}{1}}}$

${\displaystyle {\frac {p_{2}}{q_{2}}}=[2;1,2]={\frac {8}{3}}}$

${\displaystyle {\frac {p_{3}}{q_{3}}}=[2;1,2,1]={\frac {11}{4}}}$

${\displaystyle {\frac {p_{4}}{q_{4}}}=[2;1,2,1,1]={\frac {19}{7}}}$

${\displaystyle {\frac {p_{5}}{q_{5}}}=[2;1,2,1,1,4]={\frac {87}{32}}}$

${\displaystyle {\frac {p_{6}}{q_{6}}}=[2;1,2,1,1,4,1]={\frac {106}{39}}}$

${\displaystyle {\frac {p_{7}}{q_{7}}}=[2;1,2,1,1,4,1,1]={\frac {193}{71}}}$

${\displaystyle {\frac {p_{8}}{q_{8}}}=[2;1,2,1,1,4,1,1,6]={\frac {1264}{465}}}$

${\displaystyle {\frac {p_{9}}{q_{9}}}=[2;1,2,1,1,4,1,1,6,1]={\frac {1457}{536}}}$

${\displaystyle {\frac {p_{10}}{q_{10}}}=[2;1,2,1,1,4,1,1,6,1,1]={\frac {2721}{1001}}}$

${\displaystyle {\frac {p_{11}}{q_{11}}}=[2;1,2,1,1,4,1,1,6,1,1,8]={\frac {23225}{8544}}}$

${\displaystyle \dots }$

${\displaystyle {\frac {p_{20}}{q_{20}}}=[2;1,2,1,1,4,1,1,6,1,1,8,1,1,10,1,1,12,1,1,14]={\frac {410105312}{150869313}}}$

${\displaystyle \dots }$

Calculation of the decimal places

The series display is usually used to calculate the decimal places

${\displaystyle e=\sum _{k=0}^{\infty }{\frac {1}{k!}}={\frac {1}{0!}}+{\frac {1}{1!}}+{\frac {1}{2!}}+{\frac {1}{3!}}+{\frac {1}{4!}}+\dotsb =1+1+{\frac {1}{2}}+{\frac {1}{6}}+{\frac {1}{24}}+\dotsb }$

evaluated, which converges quickly. Long number arithmetic is important for the implementation so that the rounding errors do not falsify the result. A method that is also based on this formula, but does not require complex implementation, is the drip algorithm for calculating the decimal places of that A. H. J. Sale found. ${\displaystyle e}$

Development of the number of known decimal places from ${\displaystyle e}$

date	number	mathematician
1748	23	Leonhard Euler
1853	137	William Shanks
1871	205	William Shanks
1884	346	J. Marcus Boorman
1946	808	?
1949	2.010	John von Neumann (calculated on the ENIAC )
1961	100,265	Daniel Shanks and John Wrench
1981	116,000	Steve Wozniak (calculated using an Apple II )
1994	10,000,000	Robert Nemiroff and Jerry Bonnell
May 1997	18.199.978	Patrick Demichel
August 1997	20,000,000	Birger Seifert
September 1997	50,000,817	Patrick Demichel
February 1999	200,000,579	Sebastian Wedeniwski
October 1999	869.894.101	Sebastian Wedeniwski
November 21, 1999	1,250,000,000	Xavier Gourdon
July 10, 2000	2,147,483,648	Shigeru Kondo and Xavier Gourdon
July 16, 2000	3,221,225,472	Colin Martin and Xavier Gourdon
August 2, 2000	6,442,450,944	Shigeru Kondo and Xavier Gourdon
August 16, 2000	12,884,901,000	Shigeru Kondo and Xavier Gourdon
August 21, 2003	25,100,000,000	Shigeru Kondo and Xavier Gourdon
September 18, 2003	50,100,000,000	Shigeru Kondo and Xavier Gourdon
April 27, 2007	100,000,000,000	Shigeru Kondo and Steve Pagliarulo
May 6, 2009	200,000,000,000	Shigeru Kondo and Steve Pagliarulo
February 20, 2010	500,000,000,000	Alexander Yee
5th July 2010	1,000,000,000,000	Shigeru Kondo
June 24, 2015	1,400,000,000,000	Ellie Hebert
February 14, 2016	1,500,000,000,000	Ron Watkins
May 29, 2016	2,500,000,000,000	“Yoyo” - unverified calculation
29th August 2016	5,000,000,000,000	Ron Watkins
3rd January 2019	8,000,000,000,000	Gerald Hofmann
July 11, 2020	12,000,000,000,000	David Christle

literature

• Brian J. McCartin: e: The Master of All. Mathematical Intelligencer, Volume 28, 2006, No. 2, pp. 10-21. The article received the Chauvenet Prize . mathdl.maa.org
• Heinrich Dörrie : triumph of mathematics. Hundreds of famous problems from two millennia of mathematical culture . 5th edition. Physica-Verlag, Würzburg 1958.
• Leonhard Euler: Introduction to the Analysis of the Infinite . First part of the Introductio in Analysin Infinitorum . Springer Verlag, Berlin / Heidelberg / New York 1983, ISBN 3-540-12218-4 ( MR0715928 - reprint of the Berlin 1885 edition).
• Ernst Hairer , Gerhard Wanner : Analysis in historical development . Springer-Verlag, Berlin, Heidelberg 2011, ISBN 978-3-642-13766-2 .
• Konrad Knopp: Theory and Application of the Infinite Series (=  The Basic Teachings of Mathematical Sciences . Volume 2 ). 5th, corrected edition. Springer Verlag, Berlin / Göttingen / Heidelberg / New York 1964, ISBN 3-540-03138-3 ( MR0183997 ).
• Eli Maor : e: the story of a number . Princeton University Press, Princeton 1994, ISBN 978-0-691-14134-3 .
• Eli Maor: The number e: history and stories . Birkhäuser Verlag, Basel (inter alia) 1996, ISBN 3-7643-5093-8 .
• CD Olds: The simple continued fraction expansion of e . In: American Mathematical Monthly . tape 77 , 1971, p. 968-974 .
• Oskar Perron : Irrational Numbers . Reprint of the 2nd, revised edition (Berlin, 1939). 4. reviewed and added. Walter de Gruyter Verlag, Berlin 2011, ISBN 978-3-11-083604-2 , doi : 10.1515 / 9783110836042.fm .
• Oskar Perron: The theory of continued fractions - Volume II: Analytical-function-theoretical continued fractions . Reprographic reprint of the third, improved and revised edition, Stuttgart 1957. 4th revised and supplemented. Teubner Verlag, Stuttgart 1977, ISBN 3-519-02022-X .
• J. Steiner : About the greatest product of the parts or summands of every number . In: Journal for pure and applied mathematics . tape 40 , 1850, pp. 208 ( gdz.sub.uni-goettingen.de ).
• David Wells: The Lexicon of Numbers. Translated from the English by Dr. Klaus Volkert . Original title: The Penguin Dictionary of Curious and Interesting Numbers. Fischer Taschenbuch Verlag, Frankfurt / Main 1990, ISBN 3-596-10135-2 .

Web links

Commons : Euler's number  - collection of images, videos and audio files

Wiktionary: Euler's number  - explanations of meanings, word origins, synonyms, translations

• Eric W. Weisstein : e . In: MathWorld (English).
• Intuitively understandable visualization of e in an interactive Java applet
• Understandable explanation and derivation of Euler's number
• e to one million jobs at Project Gutenberg
• Detailed information and details on relevant literature (English)

References and footnotes

1. ↑ Follow A001113 in OEIS
2. ↑ Note: Euler's number is not identical to the Euler-Mascheroni constant , which in some sources has the similar-sounding name Euler's constant . ${\displaystyle \gamma }$
3. ↑ Euler: Introduction ... (§ 122) . S. 226-227 .
4. ↑ Euler: Introduction ... (§§ 123,125) . S. 91-94 .
5. ↑ ^{a b} Knopp: Theory and Application ... (§ 9) . S. 84 .
6. ↑ Euler: Introduction ... (§ 122) . S. 91 . Euler writes (according to the translation by Hermann Maser ): “In the following, for the sake of brevity, we will always use the letter for this number , so that the base of the natural or hyperbolic logarithms means […], or it should always denote the sum of the infinite series . " ${\displaystyle 2{,}718281828459\cdots }$ ${\displaystyle e}$ ${\displaystyle e}$ ${\displaystyle e}$ ${\displaystyle 1+{\frac {1}{1}}+{\frac {1}{1\cdot 2}}+{\frac {1}{1\cdot 2\cdot 3}}+{\frac {1}{1\cdot 2\cdot 3\cdot 4}}+\cdots }$
7. ^ Hans F. Ebel, Claus Bliefert, Walter Greulich: Writing and publishing in the natural sciences . 5th edition. Wiley-VCH, Weinheim, ISBN 3-527-66027-5 .
8. ↑ Paulo Ribenboim: My Numbers, My Friends: Highlights of Number Theory. Springer textbook, 2009, ISBN 978-3-540-87955-8 , p. 299.
9. ^ Richard George Stoneham: A general arithmetic construction of transcendental non-Liouville normal numbers from rational fractions . (PDF; 692 kB) In: Acta Arithmetica , 16, 1970, pp. 239-253.
10. ↑ The Stirling Formula . (PDF; 76 kB) In: James Stirling: Methodus Differentialis . 1730, p. 1.
11. ^ Ernst Hairer, Gerhard Wanner : Analysis in historical development. 2011, p. 41.
12. ^ Perron: Irrational Numbers . S. 115 .
13. ↑ Euler, p. 305.
14. ↑ Follow A073333 in OEIS
15. ↑ ^{a b} Perron: The doctrine of the continued fractions . tape II , p. 19 .
16. ↑ Perron: The doctrine of the continued fractions . tape II , p. 157 .
17. ↑ Note the connection to identity (3)!
18. ↑ Gal Raayoni et al .: The Ramanujan Machine: Automatically Generated Conjectures on Fundamental Constants . arxiv : 1907.00205 , revised version of July 23, 2019, accessed on July 28, 2019.
19. ↑ About the greatest product of the parts or summands of each number. In: Journal for pure and applied mathematics . tape 40 , 1850, pp. 208 .
20. ^ Dörrie, p. 358.
21. ↑ One can solve this problem with the methods used in the curve discussion in differential calculus .
22. ↑ Maor, p. 185.
23. ^ Wells, p. 46.
24. ↑ Olds: The simple continued fraction expansion of e. In: Amer. Math. Monthly . 1971, p. 973 .
25. ↑ See: Series A007676 in OEIS (numerator) / Series A0A007677 in OEIS (denominator).
26. ↑ AHJ Sale: The Calculation of e to Many Significant Digits . In: The Computer Journal . tape 11 , no. 2 , August 1968, p. 229–230 , doi : 10.1093 / comjnl / 11.2.229 .
27. ↑ Leonhardo Eulero : Introductio in analysin infinitorum. Volume 1, Marcus-Michaelis Bousquet and socii, Lausannæ 1748 (Latin; “2.71828182845904523536028” on books.google.de p. 90).
28. ↑ ^{a b c d e f g} Alexander J. Yee: e. February 16, 2019, accessed March 14, 2020 . 
29. ↑ Alexander J. Yee: Records set by y-cruncher. July 12, 2020, accessed on July 13, 2020 .